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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.7.49
Capitolo 10, Problema 10.7.49

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \). Notice it is an alternating series because of the factor \( (-1)^k \).
Check for absolute convergence by considering the absolute value of the terms: \( \left| \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \right| = \frac{1}{\sqrt{k^{3/2} + k}} \).
Simplify the denominator inside the square root to understand the behavior for large \( k \): \( \sqrt{k^{3/2} + k} = \sqrt{k^{3/2}(1 + k^{1/2 - 1})} = \sqrt{k^{3/2}} \sqrt{1 + k^{-1/2}} \). Since \( \sqrt{k^{3/2}} = k^{3/4} \), the term behaves like \( \frac{1}{k^{3/4}} \) for large \( k \).
Determine if the series of absolute values \( \sum \frac{1}{k^{3/4}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Here, \( p = \frac{3}{4} < 1 \), so the series does not converge absolutely.
Since the series is alternating, apply the Alternating Series Test: check if the terms \( b_k = \frac{1}{\sqrt{k^{3/2} + k}} \) decrease monotonically to zero. If they do, the series converges conditionally.

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Absolute and Conditional Convergence

Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the series converges, but the series of absolute values diverges. Distinguishing between these helps classify the behavior of alternating series.
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Percorso guidato
07:51
Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (−1)^k, as in the given problem.
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10:54
Alternating Series Test

Comparison and p-Series Tests

Comparison tests involve comparing a given series to a known benchmark series to determine convergence. The p-series test states that ∑ 1/k^p converges if p > 1 and diverges otherwise. Recognizing the dominant term in the denominator helps apply these tests effectively.
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04:30
P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)k! / (kᵏ + 3)

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Domanda del libro di testo

21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 2aₙ; a₁ = 2

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Domanda del libro di testo

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) 20 / (∛k + √k)

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Domanda del libro di testo

8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.


∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

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Domanda del libro di testo

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer

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Domanda del libro di testo

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 3 to ∞) 5 / (2 + lnk)

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